A class of quasilinear parabolic equations offorward-backward type $u_t=[\phi(u)]_{xx}$ in one space dimension is addressed, under assumptions on thenonlinear term $\phi$ which hold for a number of mathematical models in the theory of phase transitions. The notion of a three-phase solution to the Cauchy problem associated with the aforementioned equationis introduced. Then the time evolution of three-phase solutions is investigated, relying on a suitable entropy inequality satisfied by such a solution. In particular, it is proven that transitions between stable phases must satisfy certain admissibility conditions.